Abstract
A complete Riemannian manifold X with negative curvature satisfying − b 2 ⩽ K X ⩽ − a 2 < 0 for some constants a , b , is naturally mapped in the space of probability measures on the ideal boundary ∂ X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.
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